Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs.

Format:

Book

Author/Creator:

Geiss, Stefan.

Contributor:

Ylinen, Juha.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.272

Language:

English

Subjects (All):

Stochastic differential equations.

Besov spaces.

Physical Description:

1 online resource (124 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their Lpvariation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Background

1.2. Outline of the main ideas

1.3. Notation

Chapter 2. A General Factorization

2.1. The operators \C and \C^{ }

2.2. The operators \C and \C^{ } for stochastic processes

Chapter 3. Transference of SDEs

3.1. Setting

3.2. Results

Chapter 4. Anisotropic Besov Spaces on the Wiener Space

4.1. Classical Besov spaces on the Wiener space

4.2. Setting

4.3. Definition of anisotropic Besov spaces

4.4. Connection to real interpolation

4.5. The space \B_{ }^{Φ₂}

4.6. An embedding theorem for functionals of bounded variation

4.7. Examples

Chapter 5. Continuous BMO-Martingales

5.1. Continuous BMO-martingales and sliceable numbers

5.2. Fefferman's inequality and \bmo( _{2 }) spaces

5.3. Reverse Hölder inequalities

5.4. An application to BSDEs

Chapter 6. Applications to BSDEs

6.1. The setting

6.2. Stability of BSDEs with respect to perturbations of the Gaussian structure

6.3. On classes of quadratic and sub-quadratic BSDEs

6.4. Settings for the stability theorem

6.5. On the _{ }-variation of BSDEs

6.6. Applications to other types of BSDEs

Appendix A. Technical Facts

Bibliography

Index

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references and index.

ISBN:

9781470467517

1470467518

OCLC:

1275392940